Second Order Elliptic Equations and Elliptic Systems

Table of contents :
Cover
Title page
Contents
Preface to the English translation
Preface
Second Order Elliptic Equations
?² theory
Schauder theory
?^{?} theory
De Giorgi-Nash-Moser estimates
Quasilinear equations of divergence form
Krylov-Safonov estimates
Fully nonlinear elliptic equations
Second Order Elliptic Systems
?² theory for linear elliptic systems of divergence form
Schauder theory for linear elliptic systems of divergence form
?^{?} theory for linear elliptic systems of divergence form
Existence of weak solutions of nonlinear elliptic systems
Regularity for weak solutions of nonlinear elliptic systems
Sobolev spaces
Sard’s theorem
Proof of the John-Nirenberg theorem
Proof of the Stampacchia interpolation theorem
Proof of the reverse Hölder inequality
Bibliographic notes
Bibliography
Index
Back Cover

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Selected Title s i n Thi s Serie s 174 Ya-Zh e C h e n an d L a n - C h e n g W u , Secon d orde r ellipti c equation s an d ellipti c systems, 1 99 8 173 Y u . A . D a v y d o v , M . A . Lifshits , a n d N . V . S m o r o d i n a , Loca l propertie s o f distributions o f stochasti c functionals , 1 99 8 172 Ya . G . Berkovic h a n d E . M . Zhmud' , Character s o f finit e groups . Par t 1 , 1 99 8 171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 1 99 8 170 V i k t o r P r a s o l o v an d Yur i S o l o v y e v , Ellipti c function s an d ellipti c integrals , 1 99 7 169 S . K . G o d u n o v , Ordinar y differentia l equation s wit h constan t coefficient , 1 99 7 168 Junjir o N o g u c h i , Introductio n t o comple x analysis , 1 99 8 167 M a s a y a Y a m a g u t i , Masayosh i H a t a , an d J u n K i g a m i , Mathematic s o f fractals , 1 99 7 166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 1 99 7 165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n Galois theory , 1 99 7 164 E . I . G o r d o n , Nonstandar d method s i n commutativ e harmoni c analysis , 1 99 7 163 A . Ya . D o r o g o v t s e v , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko , Probability theory : Collectio n o f problems , 1 99 7 162 M . V . B o l d i n , G . I . Simonova , an d Yu . N . T y u r i n , Sign-base d method s i n linea r statistical models , 1 99 7 161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c dynamics , 1 99 7 160 V . G . Osmolovskff , Linea r an d nonlinea r perturbation s o f th e operato r div , 1 99 7 159 S . Ya . K h a v i n s o n , Bes t approximatio n b y linea r superposition s (approximat e nomography), 1 99 7 158 Hidek i Omori , Infinite-dimensiona l Li e groups , 1 99 7 157 V . B . Kolmanovski t an d L . E . Shatkhet , Contro l o f system s wit h aftereffect , 1 99 6 156 V . N . Shevchenko , Qualitativ e topic s i n intege r linea r programming , 1 99 7 155 Y u . Safaro v an d D . Vassiliev , Th e asymptoti c distributio n o f eigenvalue s o f partia l differential operators , 1 99 7 154 V . V . Prasolo v an d A . B . Sossinsky , Knots , links , braid s an d 3-manifolds . A n introduction t o th e ne w invariant s i n low-dimensiona l topology , 1 99 7 153 S . K h . A r a n s o n , G . R . B e l i t s k y , an d E . V . Zhuzhoma , Introductio n t o th e qualitative theor y o f dynamica l system s o n surfaces , 1 99 6 152 R . S . Ismagilov , Representation s o f infinite-dimensiona l groups , 1 99 6 151 S . Yu . Slavyanov , Asymptoti c solution s o f th e one-dimensiona l Schrodinge r equation , 1996 150 B . Ya . Levin , Lecture s o n entir e functions , 1 99 6 149 Takash i Sakai , Riemannia n geometry , 1 99 6 148 V l a d i m i r I . P i t e r b a r g , Asymptoti c method s i n th e theor y o f Gaussia n processe s an d fields, 1 99 6 147 S . G . Gindiki n a n d L . R . Volevich , Mixe d proble m fo r partia l differentia l equation s with quasihomogeneou s principa l part , 1 99 6 146 L . Ya . Adrianova , Introductio n t o linea r system s o f differentia l equations , 1 99 5 145 A . N . A n d r i a n o v an d V . G . Zhuravlev , Modula r form s an d Heck e operators , 1 99 5 144 O . V . Troshkin , Nontraditiona l method s i n mathematica l hydrodynamics , 1 99 5 143 V . A . M a l y s h e v an d R . A . M i n l o s , Linea r infinite-particl e operators , 1 99 5 142 N . V . Krylov , Introductio n t o th e theor y o f diffusio n processes , 1 99 5 141 A . A . D a v y d o v , Qualitativ e theor y o f contro l systems , 1 99 4 140 Aizi k I . Volpert , V i t a l y A . Volpert , an d Vladimi r A . Volpert , Travelin g wav e solutions o f paraboli c systems , 1 99 4 (Continued in the back of this publication)

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Second Orde r Elliptic Equation s and Ellipti c System s

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10.1090/mmono/174

Translations of

MATHEMATICAL MONOGRAPHS Volume 1 7 4

Second Orde r Elliptic Equation s and Ellipti c System s Ya-Zhe Che n Lan-Cheng Wu Translated b y BeiHu

•VttEHjj

American Mathematical Societ y Providence, Rhode Island Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Editorial Boar d Sun-Yung Alic e Chan g S.-Y. Chen g Tsit-Yuen La m (Chair ) Tai-Ping Li u Chung-Chun Yan g

(Second Orde r Ellipti c Equation s an d Ellipti c Systems ) Ya-Zhe Che n ( | ^ 5 E $ J T ) an d Lan-Chen g W u Copyright © 1 99 1 by Ya-Zhe Chen an d Lan-Chen g Wu Originally published i n Chinese by Science Press, Beijing, Chin a 1991 Translated fro m th e Chines e by Bei Hu ($ 3 IK) 1991 Mathematics Subject Classification. Primar y 35-01 , 35-02; Secondary 35B45, 35B65, 35J15, 35J60, 35K10. ABSTRACT. Thi s boo k i s base d o n th e authors ' lectur e note s a t th e Institut e o f Mathematic s a t Nankai Universit y durin g th e Partia l Differentia l Equation s Yea r i n 1 985 , absorbing als o th e mos t recent material s fro m th e lecture s o f experts . There ar e tw o part s o f th e book . Fo r th e Dirichle t proble m o f secon d orde r ellipti c partia l differential equations , variou s kind s o f a prior i estimat e methods , includin g th e mos t recen t tech niques, ar e rathe r completel y introduce d i n th e first part . Linear , quasilinea r an d full y nonlinea r equations ar e studied . I n th e secon d part , th e existenc e an d regularit y theorie s o f th e Dirichle t problem fo r linea r an d nonlinea r secon d orde r ellipti c partia l differentia l system s ar e introduced . This boo k choose s appropriat e materials ; i t i s a ver y goo d textboo k fo r graduat e students . This boo k ca n als o be use d a s a reference boo k fo r undergraduat e mathematic s majors , graduat e students, professor s an d scientists .

Library o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Chen, Yazhe , 1 939 [Erh chie h t' o yua n hsin g fan g ch'en g yi i t' o yua n hsin g fan g ch'en g tsu . English ] Second orde r ellipti c equation s an d ellipti c system s / Ya-Zh e Chen , Lan-Chen g W u ; translate d by Be i Hu . p. cm . — (Translation s o f mathematica l monographs , ISS N 0065-928 2 ; v. 1 74 ) Includes bibliographica l reference s an d index . ISBN 0-821 8-0970- 9 (alk . paper ) 1. Differentia l equations , Elliptic . I . Wu , Lancheng , 1 934 - . II . Title . III . Series . QA377.C44131 99 8 515'.353—dc21 97-4679 4 CIP

© 1 99 8 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . English translatio n publishe d b y th e AMS , wit h th e consen t o f Scienc e Press . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Information o n copyin g an d reprintin g ca n b e foun d i n th e bac k o f thi s volume . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 0

9 08 07 06 05 04

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Contents Preface t o th e Englis h Translatio n x

i

Preface xii

i

Part I . Secon 1 d Orde r Ellipti c Equation s Chapter 1 . L 2 Theor y 3 1. Lax-Milgra m theore m 3 2. Wea k solution s o f ellipti c equation s 4 3. Th e Predhol m Alternativ e 7 4. A maximu m principl e fo r wea k solution s 8 1 5. Regularit y fo r wea k solution s Chapter 2 . Schaude r Theor y 1. Holde r space s 2. Mollifier s 2 3. C 2 ' a estimate s fo r solution s o f potentia l equation s 2 4. Interio r Schaude r estimate s 2 5. Globa l Schaude r estimate s 3 6. A maximu m principl e fo r classica l solution s 3 7. Solvabilit y o f th e Dirichle t proble m 3

3 7 7 0 3 7 0 2 3

Chapter 3 . L p Theor y 3 7 1. Th e Marcinkiewic z interpolatio n theore m 3 7 2. A decompositio n lemm a 4 0 3. Estimate s fo r solution s o f potentia l equation s 4 1 4. Interio r W 2iP estimates 4 6 5. Globa l W 2* estimate s 4 7 2 p 6. Existenc e o f W > solution s 4 9 Chapter 4 . D e Giorgi-Nash-Mose r Estimate s 5 1. Loca l propertie s o f wea k solution s 5 2. Interio r Holde r continuit y 6 3. Globa l Holde r continuit y 6

3 3 0 3

Chapter 5 . Quasilinea r Equation s o f Divergenc e For m 6 1. Boundednes s o f wea k solution s 6

7 7

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viii CONTENT

S

2. Holde r estimate s fo r bounde d wea k solution s 6 3. Gradien t estimate s 7 4. Gradien t Holde r estimate s 7 5. Solvabilit y o f th e Dirichle t proble m 7 Chapter 6 . Krylov-Safono v Estimate s 7 1. Th e Alexandroff-Bakelman-Pucc i maximu m principl e 7 2. Harnac k inequalitie s an d interio r Holde r estimate s 8 3. Globa l Holde r estimate s 9 Chapter 7 . Full y Nonlinea r Ellipti c Equation s 9 1. Maximu m nor m an d Holde r estimate 1 s fo r solution s 0 2. Gradien t estimate s 0 1 3. Gradien t Holde r estimate s 0 4. Solvabilit y fo r quasilinea r equation s o f nondivergenc 1 1 e for m 5. Solvabilit 1 y1 fo r full y nonlinea r equation s 1 1 6. A specia l clas s o f equation s 7. 1 Genera l full y nonlinea r equation s 2 Part II . Secon1 d Orde r Ellipti c System s 2

9 2 4 6 9 9 7 6 9 0 4 7 3 5 7 2 9

Chapter 8 . L 2 Theor y fo r Linea r Ellipti c System s o f Divergenc e 1 For m 3 1 1 1. Existenc e o f wea k solution s 3 1 2. Energ 1 y estimate s an d H 2 regularit y 3 4 Chapter 9 . Schaude r Theor y fo r Linea r Ellipti c System s o f Divergenc e For m 1 3 7 1 1. Morre y an d Campanat o space s 3 7 2. Schaude r theor y 4 5 Chapter 1 0 . LP Theory fo r Linea r Ellipti c System s o f Divergenc e For 1 m 5 1. BM O space s an d th e Stampacchi a interpolatio 1 n theore m 5 2. L p theor y 5

5 5 6

Chapter 1 1 . Existenc e o f Wea k Solution s o f Nonlinea r Ellipti c System 1 s 6 1. Introductio n 6 2. Th e variationa l metho d 6

3 3 4

Chapter 1 2 . Regularit y fo r Wea k Solution s o f Nonlinea r Ellipti c System 1 s 7 3 2 1. H regularit y 7 3 2. Furthe r regularit 1 y an d counterexample s 7 8 3. Indirec t metho 1 d fo r studyin g regularit y 8 1 4. Th e revers e Holde r inequalit y an d L p estimate 1 s fo r Du 8 7 5. Direc t method 1 s fo r studyin g regularit y 9 8 6. Th e singula r se t 20 4

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CONTENTS i

x

Appendix 1 . Sobole v Space s 20 1. Wea k derivative s an d th e Sobole v spac e W kiP(£l) 20 2. Rea l1 exponen t Sobole v space s H s(Rn) 2 3. Poincare' s inequalit y 2

9 9 2 3

Appendix 2 . Sard' s Theore m 2

5

Appendix 3 . Proo f o f the John-Nirenber 1 g Theore m 2

7

Appendix 4 . Proo f o f th e Stampacchi a Interpolatio1 n Theore m 2

9

Appendix 5 . Proo f o f the Revers e Holde r Inequalit y 22

5

Bibliographic Note s 23

3

Bibliography 23

9

Index 24

5

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Preface t o th e Englis h Translatio n We are very please d tha t ou r graduat e textboo k ha s bee n chose n b y the Amer ican Mathematica l Societ y fo r translatio n int o English . I n thi s Englis h edition , some obviou s typographi c error s i n th e Chines e editio n ar e corrected ; som e name s are modifie d accordin g t o th e curren t convention . W e als o adopted th e translator' s suggestion an d adde d ver y brie f bibliographi c note s fo r eac h chapter , togethe r wit h updated references . We would like to express our wholehearted gratitud e to Professors Alic e Chang, Tsit-Yuen La m an d othe r AM S personne l fo r thei r vas t amoun t o f wor k o n thi s translation project . W e would als o like to than k th e translator , Dr . Be i Hu, fo r hi s valuable suggestions ; h e type d th e WF^i. manuscrip t fo r thi s Englis h edition . Owing to the authors' limited knowledge , errors and inappropriatenes s ar e har d to avoid ; w e welcome correction s fro m readers . Ya-Zhe Chen , Lan-Chen g W u July 9 , 1 99 7

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Preface The theor y o f secon d orde r ellipti c equation s an d system s i s fundamenta l fo r studying partia l differentia l equations , an d therefor e i t wa s liste d a s a basic cours e for graduat e student s a t th e Institut e o f Mathematic s a t Nanka i Universit y durin g the Partia l Differentia l Equation s Yea r i n 1 985 . Th e author s wer e invite d t o giv e lectures t o graduat e student s fo r thi s course . A t tha t time , th e Institut e a t Nanka i also invited man y well-know n expert s fro m aroun d th e world to giv e lectures, which provided th e cours e wit h th e mos t recen t result s i n th e area . Thi s boo k i s base d on th e authors ' lectur e notes , but w e have absorbe d als o the mos t recen t material s from th e lecture s o f th e othe r invite d experts . There ar e man y excellen t book s o n secon d orde r ellipti c partia l differentia l equations an d systems , suc h a s [GT] , [LU ] and [GQl] , listed i n th e bibliograph y o f this book . Thes e book s giv e a thoroug h introductio n i n thi s area . However , som e of these books ar e too bi g to be suitable a s textbooks fo r beginners . Th e purpos e of this boo k i s to provid e a textboo k fo r graduat e students . Thi s boo k include s bot h basic material s an d th e mos t recen t result s an d methods , s o a s t o brin g graduat e students t o th e frontie r o f thi s area . There ar e tw o part s o f th e book . Fo r th e Dirichle t proble m fo r secon d orde r elliptic partia l differentia l equations , variou s kind s o f a prior i estimat e method s are rathe r completel y introduce d i n th e firs t part . Th e Krylov-Safono v estimat e and full y nonlinea r ellipti c equations , whic h appeare d i n th e 80's , ar e introduce d in detail , bu t concisely . I n th e secon d part , the existenc e an d regularit y theories o f linear an d nonlinea r secon d order ellipti c partial differential system s ar e introduced . Basic facts abou t Sobole v spaces are given in Appendix 1 . I n order t o emphasize th e main theme , th e proof s o f som e theorems , suc h a s th e Stampacchi a interpolatio n theorem an d th e revers e Holde r inequality , ar e include d i n th e Appendix . Owing to th e authors ' limite d knowledge , error s ar e har d t o avoid ; w e welcome suggestions fro m readers . Under th e leadershi p o f Professo r Li-Shan g Jian g (Hr^ L inj), th e partia l dif ferential equatio n semina r a t Pekin g Universit y playe d a n importan t rol e i n thi s book. W e would lik e to expres s ou r dee p gratitude t o Professo r Li-Shan g Jian g an d the semina r participant s fo r thei r contributions . W e would als o lik e to expres s ou r wholehearted gratitud e t o Professo r Guang-Li e Wan g (ZE^/fvl ) fro m Jili n Univer sity, wh o rea d th e draf t o f thi s boo k an d mad e valuabl e suggestions . Ya-Zhe Chen , Lan-Chen g W u May 20 , 1 99 0

xii i

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Part I

Second Orde r Ellipti c Equation s

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10.1090/mmono/174/01

CHAPTER 1

L2 Theor y The stud y o f solvability o f the Dirichle t proble m fo r ellipti c equations i s one of the central topics of this book. Th e introduction o f Sobolev spaces (cf . Appendi x 1 ) provides a n effectiv e too l fo r th e study . I n Sobole v spaces , w e ca n see k solution s in a mor e genera l clas s o f functions ; thi s make s solvabilit y problem s muc h easier . Solutions of this kind ar e often referre d t o as weak solutions o r generalized solutions. Of course, in order t o obtain th e existence of a classical solution, w e must stud y th e smoothness o f wea k solutions . Suc h a proble m i s called th e regularit y problem . I n §4 of this chapte r an d nex t chapter , w e will explore the basi c method s fo r studyin g the regularit y problem . 1. Lax-Milgra m theore m Let H b e a rea l Hilber t spac e an d H' it s dua l space . Denot e b y (• , •) the dua l product betwee n H an d H'. Definition 1 .1 . Le t a(u,v) b e a bilinea r for m o n th e Hilber t spac e H, (i) a(u, v) i s said t o b e bounded, if there exist s M > 0 such tha t (1.1) |a(u,t;)

| ^ M | M | H | M | H , VU,V

G H.

(ii) a(u,v) i s said t o b e coercive, if ther e exist s S > 0 such tha t (1.2) a(u,u)>5\\u\\

2

H,

VueH.

Theorem 1 . 1 (Lax-Milgra m theorem) . Let a(u,v) be a bounded, coercive bilinear form on H. Then for any f G H', there exists a unique u G H such that (1.3) a(u,v

) = (/,v) , VvGH,

and

(1-4) \\u\\

H.

Proof. Fo r eac h fixe d u G H, a(u, • ) is a bounde d linea r functiona l o n H, an d therefore ther e exist s a uniqu e Au G H' suc h tha t (1.5) a(u,v)

G H,

= (Au 9v), \fv

and \\Au\\w ^ M\\u\\

H.

3

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1. L 2 THEOR Y

4

It ca n easily b e verified tha t A i s linear. B y the coerciveness, w e have ||AU||/J'IMIH ^ > a(u,u) ^ S\\u\\jj, and therefor e (1.6) \\Au\\

H,

Z S\\u\\ H.

It follow s tha t A i s one to one. W e will prov e tha t th e range o f A i s R(A) = H'. First, w e show tha t R(A) i s closed. I f Au n— > v, then b y (1.6) \\un - u m\\H ^ ^\\Au n - Au

m\\H>\

thus {u n} i s a Cauchy sequence . Fro m th e completeness w e conclude tha t it s limit u E H exists . B y the continuit y o f A, w e derive tha t Au n —> Au, an d therefor e v = Au £ R(A). Thi s show s tha t R(A) i s a closed subspac e o f H'. I f R(A) ^ # ' , then i t follows from th e orthogonal decomposition theore m tha t ther e exists v' G H' such tha t v' ^ 0 and v' _ L R(A). Sinc e i 7 is reflexive, ther e exist s v G H suc h tha t (*>'» ')H' = (' v)> Usin g th e coerciveness, w e ge t 0 = (v',Av) Hf =

(Av,v) ^ S\\v\\

2 H

> 0.

This contradictio n implie s tha t R(A) = H\ an d therefore ther e i s a uniqu e u suc h that Au = f. Fro m (1 .5 ) and (1.6) we immediately deduc e (1 .3 ) and (1.4). D 2. Wea k solution s o f ellipti c equation s Let £1 b e an ope n domai n i n R n . Fo r simplicity, w e shall alway s assum e tha t n ^ 3 . In this chapte r w e shall conside r ellipti c equation s o f divergence for m o n ft: (2.1) Lu

= -Dj(a ijDiU +

d?u) + (VDiU + cu) = / + A f •

We us e the summation conventio n throughou t thi s book ; repeate d subscript s an d superscripts ar e summed fro m 1 through n , and Di = —— . A s for the operator L , OXi

we shall assum e throughou t thi s chapte r tha t aij eL°°(ft) , and tha t ther e exis t positiv e constant s A , A such tha t 2

(2.2) A|£|

^ ( x )^ « ; A|£|2, V£eR

n

,sefi,

nn

(2-3) £

iMU-cn ) + £ H i=l i=

rfi

H™ + IMIL""^) < A "

l

We shal l denot e th e Sobolev spac e W fc'2(ft) b y H k(tt). Fo r u,v G #H^)>

we s e t

a{u, v) = / | (a ljDiU + d ju)DjV + (b lDiU + cu) v [da;, t/ ri

From th e proof o f Lemma 2. 1 below w e shall see that th e integrals ar e well define d in the above equality .

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2. WEA K SOLUTION S O F ELLIPTI C EQUATION S 5

Definition 2.1 . Fo r T £ H~ l{Si) (th e dua l spac e o f H%{0,)) and g £ H we sa y tha t u £ H 1 (ft) i s a weak solution o f th e Dirichle t proble m

1

^),

f Lu = T i n ft, Iu = g o

n cK2,

if i t satisfie s

ju-seffoHfi) Lemma 2.1 . Le t £h e assumptions (2.2 ) anc i (2.3 ) 6 e in /orce, and let Q be an open bounded domain in R n . Then a(u,v) is a bounded bilinear form on HQ(Q). Proof. Usin g Holder' s inequalit y an d (2.2) , we obtai n / a lJDiuDjvdx ^ A | | u | | H i ( n ) | | t ; | | f f i ( n ) . I Jn Applying Holder' s inequality , (2.3 ) an d th e embeddin g theorem , w e deriv e

\i

djuDjvdx\ ^

di

2ll

l^ n ( n )IHlLa*(n)IPj«IUa(n)

3

< CMW\\ Hi(n)\MHl(n), I cuvdx I Jn

^ ll

C

ll L n / 2 (fi ) ll^l^ 2 * (O) IMIl*2* («)

< CA||u||

Hi(n)||t;||Hi(n),

where 2 * = 2n/(n — 2), an d th e constan t C depend s onl y o n n. Th e remainin g terms ca n b e estimate d i n a simila r way . Thu s w e obtai n

(2-6) K«,«)I^C'A||«||

if i (n) ||t;|| if i (n) .

•

Remark. Fo r eac h fixed u £ if 1 (Q), a(iz , •) i s a bounde d linea r functiona l o n HQ(Q). Analogousl y t o th e proo f o f Theore m 1 .1 , ther e exist s a bounde d linea r operator L : if1 (ft ) - * H~ 1 (fl) suc h tha t (2.7) a(u,v)

= (Lu,v), Vu£H

1

(n), v

£ H^(Q).

From no w o n w e shall no t distinguis h th e operato r L fro m th e operato r L give n i n (2.1). Lemma 2.2 . Let the assumptions (2.2 ) and (2.3 ) be in force, and let Q, be an open bounded domain in R n . Then there exists ~j2 > 0 such that a(u , v) +fi(u, v)^^ is coercive on HQ(Q) forfjL ^ Ji, where (•, -)L 2{SI) denotes the inner product on L 2(Q). In orde r t o prov e th e lemma , w e nee d th e followin g fact : fo r / £ L p(fL) an d arbitrary positiv e e > 0, ther e exist s a decompositio n f = f 1 + f 2 suc h tha t (2.8) Wf2\\

L'(Q)

< e, s*j>\h{x)\K,

where K i s sufficientl y large . Proof o f Lemm a 2.2 . Fo r an y e > 0 , ther e ar e decomposition s 6* =&*+&*, c f = d i + 4

>c

= ci+c 2,

such tha t

E U^H^ W + E I|4|IL"(Q ) + l|C2ll L»/a(n) ^ £>

E ii»iiu-(ii)+ E II4HL-(Q) + IMIL~(Q) < w . Set a2(u, v ) ~ \

(a ljD{U + d J2u)DjV + {b\DiU + C2ix) v [dx,

ai(ix,v) = a(u,v ) — a2(it,v) . Using th e positive-definitenes s conditio n (2.2 ) an d a simila r computatio n a s i n Lemma 2.1 , we easily obtai n a2(u,u) ^

(A-Ce)H^i

(n)

.

We fix e suc h tha t C e = A/4 . Fo r ai(tz,u) , w e hav e |ai(u,u)| ^

CTTO 0 such that for JJL > ~jl, the nonhomogeneous Dirichlet problem

{

Lu + uu = T , 1

u-^GfToHn)

/ms a unique weak solution.

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7

3. TH E FREDHOL M ALTERNATIV E

Proof. B y th e definitio n o f a wea k solution , th e bilinea r for m correspondin g to (2.9 ) i s give n b y a(u,v) + /i(ix , V)L 2 (Q)- A wea k solutio n satisfie s f a(u,v)+/x(u,v) o = /,

M + e + F0 - (k - I) We tak e fco— / = ( 1 — T))(M + e + Fo), wher e7 7 i s a small positiv e constan t t o be determined. The n

|A(* 0 )| 1 / 2 *^c[log-l . L^ J It i s clear tha t ther e exist s7 7 > 0 so tha t (4.1 5 ) i s valid. B y (4.1 7) , ess sup u ^ s u p u + + (1 - rj){M + e + F0 ) + CF 0 |ft| ( 1

/n)

~(1

/p)

.

Since M = ess sup u — sup u+ an d £ is arbitrary, w e conclude tha t dQ

esssupu ^ supu + + C F 0 | f i | ( 1 / n ) ~ ( 1 / p ) . The proo f i s complete . • Remark. I f the assumptio n (2.2 ) i s replaced b y th e stronge r assumptio n

(2-2)' £

IIVH^n ) + £ || n, then th e constan t C in the conclusio n (4.9 ) o f Theorem 4. 2 depend s only o n n,p , A/A an d ft an d i s independent o f 6*, ef, c and th e lowe r boun d fo r |H| .

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5. REGULARIT Y FO R WEA K SOLUTION S

13

In fact , fo r an y e > 0, w e can prov e tha t / \d?ipDjip + \Pu>Dj& + ap 2\dx\ 3 nI 3 J I

(4.24) U

For example , th e first ter m o n th e left-han d sid e o f (4.24 ) ca n b e estimate d a s follows: / d^ipDjipdx

^ II^IILHI^IIL^/(P-

I Jn

2 )II^^IU 2

Using (2.2) ' an d th e embeddin g theorem , w e then deriv e (4.24) . Thu s th e constan t C depend s onl y o n n,p , — an d Q. Ther e i s n o chang e i n othe r part s o f th e proof . A

The detail s ar e lef t a s exercise s fo r th e reader . Theorem 4.3 . Let the assumptions (2.1 ) , (2.2 ) be in force, and let c - Did 1 ^ 0 (in

the sense of £&'(Q)),

where Q is an open bounded domain on which the Sobolev embedding theorem is valid. Then there exists a unique weak solution to the Dirichlet problem, and (4-25) IMIffX(n

) < C(\\T\\ H-HQ) +

\\g\\

Hx(a)).

Proof. Withou t los s o f generality, w e may assum e tha t g = 0 (cf . th e proo f o f Theorem 2.3) . I f T = 0 , the zer o solution i s the onl y solution fo r (2.4) , by th e wea k maximum principl e (Theore m 4.1 ) . B y Theore m 3.2 , ther e exist s a uniqu e wea k solution u £ HQ(CI) fo r (2.4) , fo r an y T e H~ 1 (QI). I t follow s tha t L i s invertible , i.e., Nltfi(n) ^ C\\Lu\\ H-i(cl), Vu

e H^(Q).

From thi s inequality , (4.25 ) follows . D

5. Regularit y fo r wea k solution s For simplicity , w e consider onl y th e followin g typ e o f equation : (5.1) Lu

= -Dj(a ijDiu) +

VDiU + cu = f.

Theorem 5.1 . Let the assumption (2.2 ) be in force, and let a %i e W ltOC(Q), b\c e L°°(f2) , / G L2(ft). If u G if1 (ft ) is a weak solution of (5.1 ) , then for any ft' C C fi (i.e., ft' is a compact subset of SI) we have u E H 2(Qf), and (5-2) l|u||ff»2{fl). If a* J', 6 l, c and / ar e infinitely man y times differentiable, the n for an y k we have u G Wz * ' (ft) , an d therefore , b y th e Sobole v embeddin g theorem , u G C°°(ft) .

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10.1090/mmono/174/02

CHAPTER 2

Schauder Theor y The regularit y theor y fo r classica l solution s o f a secon d orde r ellipti c equa tion wa s firs t establishe d b y Schauder . A s fa r a s classica l solution s ar e concerned , the Schaude r theor y i s quit e complete . I t provide s a foundatio n fo r th e stud y o f nonlinear ellipti c equations . Here w e us e a ne w metho d introduce d b y Trudinger ; hi s proo f avoide d th e tedious computation s o f potentials .

1. Holde r space s In the process o f studying potentia l equations , it i s very inconvenient t o discus s continuous differentiabilit y alon e — th e result s ar e no t ric h an d plentiful . I t i s therefore necessar y t o introduc e th e concep t o f Holde r continuity , whic h ca n b e viewed a s fractiona l differentiatio n i n a certai n sense . Definition 1 .1 . Le t ft C M n, le t u(x) b e define d o n ft, an d le t XQ e f2 . If , fo r some 0 < a < 1 ,

then u i s said t o b e Holder continuous at xo with respect to Q, with exponent a. I f a — 1 in th e abov e definition , the n u i s said t o b e Lipschitz continuous at x$. We denote by C k{Sl) = C k,0(fl) th e space of functions whic h are /c-times continuously differentiat e o n Q. Fo r Holde r continuity , w e introduc e th e correspondin g class o f spaces, usuall y referre d t o a s Holde r spaces . For 0 < a ^ 1 , introduce th e semi-norm s (1.2) M o , 0 ;

« = [u] 0-Q = SU p \u(x)\,

(1.3) M o , a ;

Q = H a j Q = SU p H^

Q

(1.4) Mfc (1.5) Mfc.ajf

fo;n

Q[u]Q],

= M* ; n = X I I ^ o j f t * M=fc

i = Yl l D"u] 17

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2. S C H A U D E R T H E O R Y

18

where k i s a positiv e integer , v = {y\,v 2,- — ,^ n ) i s a multi-index , Vi ^ 0 (i = n

1,2,-•• ,n) , \v\ = ^2vi, an d 2 =1

D"u =

dx?~>dx%>

For simplicity , w e ofte n us e [D ku]a]n t o represen t V ^ [JD^UJQ-JQ . Definition 1 .2 . W e us e C fc ' a (Q) ( 0 < a ^ 1 ) t o denot e th e spac e consistin g of al l function s i n C k($t) satisfyin g [u]k^n < oo. Introduc e th e nor m o n C k(Q) k

(1.6) Mfc

;n=

5^M

TO;n ,

m=0

and th e nor m o n C fc ' a (Q) (1.7) |w|fc,a

;n

= Mfc ;n H - Mfc,a;n.

It i s no t har d t o verif y tha t bot h C k(Vt) an d C fc ' a (Q) ar e Banac h spaces . W e often dro p th e subscrip t Q in th e norm s i f ther e i s n o confusion . For th e produc t o f tw o functions , th e Holde r nor m satisfie s Lemma 1 .1 . For u,v G Ca (ft) ( 0 < a < 1 ) , (1.8) [uv]

a

^

[u] 0[v]a +

MaH o^

MaMa -

The proo f i s left t o th e reader . One o f th e mos t importan t propertie s i n Holde r space s i s th e interpolatio n inequality, whic h make s i t possibl e t o stud y onl y th e mos t importan t ter m i n de riving a n a prior i estimate , an d thu s simplifie s th e proof . Her e w e shal l us e th e compactness argumen t fo r it s proof . T h e o r e m 1 .2 . Let Q, be a bounded domain, and u € C Then for any e > 0, (1.9) [u]

2,a

(Ct) ( 0 < a ^ 1 ) .

2^e[u]2,a+C£\u\0,

(1.10) [u]i

^ e[u] 2,a + C e\u\o,

where C E depends on n , a, £1 , in addition to the dependence on e. All the norms and semi-norms are defined on Q. Proof. W e prov e onl y (1 .9) ; th e proo f fo r (1 .1 0 ) i s similar . I f ther e i s n o constant C e suc h that (1 .9 ) i s valid for al l functions i n C 2 , a (r2), then fo r an yA T > 0 , there exist s u^ suc h tha t (1.11) [u

N}2

> e[u N]2,a + N\u N\0 (N

= 1 ,2 , • • •) .

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1. H O L D E R SPACE S

19

Using th e homogeneit y o f th e inequality , w e can assum e withou t los s o f generalit y that \UN\2 = 1 , for otherwis e w e ca n replac e u^ b y UN/\UN\2- Fro m (1 .1 1 ) , (1.12) [u

N)2,a

< ~ , \u

N\o

= 1 ,2 , • • •).

< — {N

Since UN is uniforml y bounde d i n C 2 , a (Q), ther e exist s a subsequenc e u^ k whic h converges t o som e u i n C 2(S1 ), by th e Ascoli-Arzel a theorem . However , th e secon d inequality i n (1 .1 2 ) implie s tha t u^ converge s t o 0 uniformly o n Q . I t follow s tha t u = 0 , whic h contradict s \UN\2 — 1- Th e proo f o f (1 .9 ) i s complete . • Definition 1 .3 . A domai n f i i s sai d t o satisf y a cone property, i f ther e i s a finite con e V suc h tha t fo r an y x G fi, ther e exist s a con e congruen t t o V wit h vertex x an d containe d completel y i n Q,. Theorem 1 .3 . Suppose that Q satisfies a cone property with h the height of the cone. Then for any 0 < e ^ h, we have a

(1.13) [u]

(1.14) M i < e

2^e

[u}2}a +

1 +a

(j ^\u\

0j

M2,a + j H o ,

where the constant C depends only on n and the solid angle of opening of the cone. Proof. Le t V\ b e a con e wit h th e sam e soli d angl e o f openin g an d heigh t 1 . Then Theore m 1 . 2 implie s tha t (1.15) [ix]

2;vi

^ [ " h a ; * + C\u\ 0lVl fo

r u G C2 ' a (Fx),

where C depend s onl y o n th e soli d angl e o f opening . Fo r u G C2'a(VE) (V e share s the sam e soli d angl e o f openin g a s V\ , wit h heigh t e an d verte x a t th e origin) , w e use th e chang e o f variables y = x/e, u(y) = u{ey) = u(x). The n u G C2 ' a (Vi), an d (1.15) i s valid fo r u. I f w e rewrite thi s inequalit y i n term s o f the origina l variables , then Q M2;Ve ^ £ aM2,a;V£ + " ^ No-,Ve. For an y x G ft an d e ^ ft, ther e exist s a con e V e wit h verte x x an d V e C O . Thus C

C

Since a ; is arbitrary , M2;Q ^ f a M 2 , a ; f l + ~o Mofr-

The proo f o f (1 .1 4 ) i s similar. •

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20

2. SCHAUDE R THEOR Y

2. Mollifier s An equivalent Holde r nor m o f a function ca n b e introduced usin g derivative s of its mollifie d function . Thi s wil l reduce th e Holde r estimate s fo r solution s o f ellipti c equations t o estimatin g th e derivatives , an d therefor e wil l simplify th e proof . Let p £ C£°(R n ), p(x) ^ 0 with suppor t spt{p } C #i(0 ) an d (2.1) /

= 1.

p(x)dx

Such a functio n i s called a mollifier. On e ca n take , fo r example ,

{

k exp { —pr—- [ fo 0 fo

r \x\ < 1 ,

r \x\ ^ 0 ,

where k i s chosen s o tha t (2.1 ) i s valid . Definition 2 . 1 . Fo r u £ L/ oc (R n ) an d p{x) a mollifier , th e functio n

) = r~ n f p(^l)

(2.2) 5(x,r

u(y)dy

is said t o b e a mollified function o f u(x). Lemma 2.1 . Le i ^ £ C(W l). Then pact set to u(x) as r —» • 0 , an d (2.3) su

S(x,r ) converges uniformly on any com-

p |S| ^ su p |w|, k

(2.4) \D

k

u(x,r)\^Cr- su

p |u | (Jf

e = 0,1 ,2 , • • • ),

J3 T (x )

where D denotes the gradient in (x , r) an d C depends only on n, fc and the mollifier PProof. B y (2.1 ) , U(X,T)-U(X)

=

r~

n

/ p ( )(^(y)-w(a:))d2

jRn

(2.5) r =/

/

T

p(n)(u(x

- rrj) - u(x))drj,

^Bi(O)

from whic h i t i s no t difficul t t o deriv e tha t 5(x,r ) converge s uniforml y o n an y compact se t t o u{x) a s r - > 0 . T o prov e (2.4) , w e first notic e that , b y induction , Dku(x,T) =

r- n-k f

P

k(^^)u(y)dy,

where P k £ Co°(R n ), sp t P k C £?i(0) . Lettin g z = (x — y)/r i n th e abov e equality , we ge t \Dku(x,r)\ ^

r~

k

k(z)u(x

^ r~

k

su p \u\ / \P

\[ P

BT(x) JR

n

-

Tz)dz\ k{z)\dz

^

Cr~

k

su p \u\. D

B

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T{x)

2. MOLLIFIER S

21

Next w e shal l discus s th e relationshi p betwee n th e derivative s o f 5(x , r) an d the Holde r nor m o f u{x). Lemma 2.2 . Let u G C£c (R n ) ( 0 < a ^ 1 ) . Then ) - u{x)\ ^ r a # * [ u ; J5 r (z)],

(2.6) |5(s,r k

u(x, r)\ ^ CT"- kH«[w, B

(2.7) \D

T(x)]

k = 1 ,2 , • •. ,

for

where C depends only on n , a,A; and the mollifier p. Proof. Fro m (2.5 ) w e easil y deriv e (2.6) . T o prov e (2.7) , w e denot e b y / ? = (/30,/3) th e n + 1 dimensional multi-inde x wit h |/3 | = fc, D& = D!*>D%. I f / 3 = 0 , then w e differentiate (2.5 ) wit h respec t t o r t o obtai n D^(x,r) =

T - " - * J^{^-iy = r~

k

/ Pp(z)(u(x

u{y)-u{x))dy

- rz) - u(x))dx,

where Pp i s a functio n wit h it s suppor t i n B\(0). Fro difficult t o deriv e (2.7) . I f 0 ^ 0 , the n D?U(X,T)

= r~ n J n =

r

m thi s equalit y i t i s no t

D^U^)]u(y)dy

-£^[p(^)](u(y)-u(x))rfy + ( - l ) ^ l r - » u ( x ) f D*DP\p(Z^)\dy.

The secon d ter m o n th e right-han d sid e o f th e abov e formul a vanishes , b y th e divergence theorem . B y estimatin g th e first term , w e derive (2.7) . D Lemma 2.3 . Let u G C(Rn). If sup r

1

~a\Du(y,r)\
8)

P*C]

) = l forxGB

o + ( 1 - rr/T [I? f c C]a ^

(1

_^

Ti,, f c

,

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4. INTERIO R SCHAUDE R ESTIMATE S 2

9

where C — C(n,k), 0 < T < 1 . Suc h a functio n ca n be constructe d b y usin g a mollifier. Le t v = £u . The n v E C%,a(BR), an d Lv = C/ + [-a ijDi:iC + VDiQu - 2a

ij

Di(Dju.

In order to apply Lemma 4.2, we need to estimate the Holder norm of the right-han d side o f the above equation . W e firs t estimat e on e of the terms: A

[a^DiCDju]* ^

a {[DC]aNi

< r f [u]l

+ PC]oMi, a + [^C]oMi } fil

t

~alttM

^\ ( 1 - r) 1 +«fl 1 + t t ( l - r ) f l j ' Using th e interpolation inequalit y (Theore m 1 .3) , we obtain, fo r any £ G (0,1),

[««ACi>j«]. < cjel^uja + C ( 1 _ r ) ( i " i ; + 1 J ^ } . Similarly, b y Lemmas 4. 2 and 1.1, Theorem 1 . 3 and (4.8), [D2v]a,BR ^

(?

{_ +

^ [/]o;B

fl

+ [f]a;B

R

+

e[D

2

uUBR

(l_ r)(2/a)+li?2+alUlo^|-

Notice tha t £(x ) = 1 for x € B T R - Usin g th e product rul e fo r derivatives an d the interpolation inequalitie s (1 .1 3) , (1 -1 4) , we obtain {D2u]a.,BTR ^

2

cUf) + {1

a,BR+e[D

u]a,BR 1 fi(a/a)

- -ftNo.-B. + « ( 2 / Q ) + 1 - Q [ / ] O ; B B ] } •

-r)W*+W°»> [

Let dB+\dB! and |(/?iv — ^IO-B + "~ * 0 a

s i V -» oo .

The boundar y valu e proble m j -a

lj

DijUN +

[ UN — ^PN o

WDiUx + C-UJ V = / i

n B^",

n $£?+

2,a

has a solutio n ttj v G C (B1 ) , b y Theore m 7.1 . Usin g th e interio r Schaude r estimate an d th e maximum principle , w e see that a subsequenc e {?Mr fe} o f {u^r} converges uniforml y o n B^ t o som e functio n 2 , an d fo r an y Q' C C -B^ 1 ", {^jv fc} converges i n C 2(f2 ) to u. Lettin g N = iVf c—> o o in (7.7) , we obtain -aijDijU + u= uo

WDiU + cu = f i

n J5^,

n 95^".

It follow s tha t S = w in B*. Applyin g Lemm a 5. 2 to (7.7) , we ge t [D2uN]a;Bt/2